# Effect of Tree Size Heterogeneity on the Overall Growth Trend of Trees in Coniferous Forests of the Tibetan Plateau

^{1}

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^{†}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Study Area and Data Collection

#### 2.2. Quantification of Tree Radial Growth Pattern and Its Overall Variability

_{r}is the relatively stable respiration cost required to produce a unit of tissue, and m

_{r}is the variable maintenance respiration rate per unit of tissue [26]. The value of T is related to the thermodynamic significance of respiration and ranges between 0 and g

_{r}/m

_{r}[15]. To counteract natural degradation (entropy increase), organisms must continuously use negative entropy to maintain the complexity, variety, and order of their components. During time T, the growth energy proportional to g

_{r}decreases the entropy of a new unit tissue relative to that of their free precursor monomers [27]. Meanwhile, the maintenance energy, which is proportional to Tm

_{r}, maintains a low entropy state of an existing unit tissue and indicates its entropy accumulation during this time. Assuming the old and new units of tissue are identical, the synthesis of a new unit of tissue is possible only if Tm

_{r}is less than g

_{r}; that is, T < g

_{r}/m

_{r}.

_{r}/m

_{r}provide us with upper and lower boundaries for f(r). Usually, b is considered equal to 0.75 [28,29], but some evidence suggests that it may be equal to 0.85 [30]. Since the following results support the former, we give here only two growth boundaries at b = 0.75:

_{r}/m

_{r}), the closer f(r) is to Equation (3) [15]. We termed Equations (2) and (3) as the thermodynamic lower (IGMR-L) and upper (IGMR-U) boundaries of the IGMR. It is worth noting that the mathematical form of extending the OGM (T → 0 and b = 0.75) [17] to the tree ring scale (OGMR) is consistent with Equation (3).

_{r}/m

_{r}× (2b + 2)/(1 − b). However, the TGT for Equation (2) is 32/3 × g

_{r}/m

_{r}. It is also likely that the pattern that tree radial growth follows will shift from Equation (2) to Equation (3) over time, therefore ensuring that the TGT of tree radial growth is consistent with that of biomass growth. We speculated that the k value is closer to 3/4 due to T → g

_{r}/m

_{r}. This also means that the actual f(r) distributed along the r gradient will be more distributed below Equation (3), as shown in Figure 2. Note that the other case is when b = 0.85 and T → g

_{r}/m

_{r}, k = 0.85.

_{i}) and f(r

_{i})

_{HBGT}denote the current average growth rate over the past five years and estimate the historical best growth rate, respectively, for the tree i. When the OVG is greater than 0, it means that the overall growth trend of the trees is better than the historical one; otherwise, this trend may decline or maintain the status quo.

#### 2.3. Data Processing and Analysis

_{c}), age (L), and average growth rate over the past five years (f(r)

_{c}), for each tree core. Additionally, we extracted the average tree ring growth rate (f(r)

_{m}) for trees of the same species within each site. To ensure the robustness of our findings, we only analyzed chronologies with a minimum of 30 samples. We assumed that tree TGT is the 95th percentile of the L values of all trees of the same species within that site. However, this estimation tends to overestimate TGT for most trees because of growth heterogeneity. Mathematically, we can still assume that TGT is accurate, but f(r)

_{m}is overestimated. Therefore, the actual relationship between the normalized tree diameter (r/R) and the normalized growth rate (f(r)

_{c}/f(r)

_{m}) should be lower than the normalized Equation (4). Note that R here is equal to TGT × f(r)

_{m}. The normalized Equation (4) is obtained by assuming R = 1 and m

_{r}/g

_{r}= 1. Afterwards, we calculated the coefficient of variation of the tree radius (CVR) at each site and tested whether the tree radial growth trajectories conformed to Equation (4) (where k = 0.75). Based on the test results, we estimated the HBGT and CVR of each site using Equations (4) and (5).

## 3. Results

#### 3.1. Tree Radial Growth Follows the IGMR-U

#### 3.2. Effect of Tree Size Heterogeneity on Overall Growth Variability

#### 3.3. Tree Radial Growth Assessment

## 4. Discussion

#### 4.1. Tree Size or Radius Constrains Its Radial Growth and Growth-Climate Sensitivity

_{r}/g

_{r}, and the potential maximum size or radius, where environmental and resource intakes could significantly affect the m

_{r}and the maximum size or radius. Some evidence supports this conclusion. For example, age effects and sampling strategy affect the accuracy of a tree growth assessment and its climate response [20,37,38]. Moreover, changes in tree physiological status [34] result in different climate-growth relationships [37,38] and inevitably feed back into the size-to-growth constraint [17,39,40]. In fact, size has a greater effect than cellular senescence on age-related declines in relative growth and net assimilation rates [41]. Our model highlights that tree size, specifically the radius, determines the radial growth trend and climate sensitivity (Figure 3D). That is, the variation coefficient of the radial growth rate still shows a single-peaked pattern on the radius gradient.

#### 4.2. Limited Influence of Climate on Tree Size Heterogeneity of Subalpine Forests

#### 4.3. Forest Range Response to Tree Size Heterogeneity and Climate Change

#### 4.4. Tree Growth Assessment Based on Generalized Metabolic Growth Theory

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Glossary

Symbol | Meaning | Unit |

f(r) | tree ring growth rate | mm/y |

T | The formation time of unit tissue is primarily controlled genetically and by physiological activities, with the intrinsic or developmental growth rate independent of organism size | y |

R | Tree maximum radius | mm |

r | Tree current radius | mm |

m_{r} | Rate of maintenance respiration per unit of tissue | mg g^{−1} y^{−1} |

g_{r} | Cost of respiration needed to produce a unit of tissue | mg g^{−1} |

b | Metabolic exponent, taken here as 0.75 | 1 |

TGT | Total growth time | y |

f(r)_{HGBT} | Growth rate of historical best growth trajectory | mm/y |

OVG | overall average growth variability | 1 |

c(r_{i}) | current average growth rate over the past five years, for tree i. | mm/y |

f(r_{i})_{HBGT} | Estimated historical best growth rate for tree i | mm/y |

f(r)_{c} | average growth rate over the past five years | mm/y |

r_{c} | Statistical current diameter | mm |

L | Statistical tree age | y |

f(r)_{m} | Statistical average tree ring growth rate | mm/y |

k | Pending parameter | 1 |

CVR | coefficient of variation of the tree radius | 1 |

## Appendix A. Iterative Growth Model (IGM)

_{r}is the cost of respiration needed to produce a new unit of tissue; and m

_{r}is a unit of tissue’s rate of maintenance respiration (per unit of time). Generally, g

_{r}is stable, and m

_{r}is sensitive to the environment and is driven by temperature, with its trend following the Arrhenius equation. Mathematically, Equation (A1) highlights a growth iterative mechanism. Namely, growth can be described as a series of spontaneously iterated feedbacks, each of length T. At each iteration, the organism moves from the initial biomass m

_{0}(slightly larger than the threshold biomass for growth, o) and approaches the final mass M. Thus, we refer to Equation (A1) as an iterative growth model (IGM). Mathematically, classical metabolic growth theory, also known as the ontogenetic growth model (OGM), is a special case of IGM at b = 0.75 and T → 0.

_{r}/m

_{r}. We derive this from the thermodynamic significance of respiration. To counteract natural degradation (entropy increase), organisms must continuously use negative entropy to maintain the complexity, variety, and order of their components. Usually, organisms obtain useful energy (e.g., chemical energy stored in photosynthetic products or food) from the environment and return equivalent amounts of energy to the environment in less useful forms, such as dissipated energy or heat. In this process, energy provides negative entropy or the required order to organisms. Thus, from an entropy perspective, during time T, the growth energy proportional to g

_{r}decreases the entropy of a new unit of tissue relative to that of their free precursor monomers [27], causing free monomers to achieve an appropriate ordered state. At the same time, the maintenance energy proportional to Tm

_{r}contributes the negative entropy to maintain the low entropy state of a unit of old tissue, and m

_{r}and Tm

_{r}are also proportional to the entropy increase rate and entropy accumulation of a unit of old tissue during time T, respectively. Assuming there is no difference between the new and old units of tissue, the new unit tissue can be synthesized only when Tm

_{r}must be less than g

_{r}, i.e., T < g

_{r}/m

_{r}. When T → 0 and g

_{r}/m

_{r}, integrating or iterating Equation (A1) will produce two smooth functions driven by time (t), i.e., the Richards and Gompertz equations [15].

^{b}

^{−1}× m

_{0}

^{1}

^{−}

^{b}, r = m

_{r}/g

_{r}(1 − b), m

_{0}is the first biomass observed, and n is the number of iterations and is equal to t × m

_{r}/g

_{r}. These results indicate that the actual growth dynamics lie somewhere between these equations (Equations (A2) and (A3)) and may not be an explicit analytic solution in most cases.

_{r}/m

_{r}× (2b + 2)/(1 − b) represents the total growth time. The basis for this equation is an integral transform from f(m) to M [15].

## Appendix B

**Figure A1.**Distribution of r/R over different CVR intervals. CVR: the coefficient of variation of the tree radius.

**Figure A2.**Correlation of the precipitation of the coldest quarter with the coefficient of variation of the tree radius (CVR) at different elevation ranges. The black lines (

**A**,

**B**) are linear regression.

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**Figure 2.**Classical metabolic growth theory (OGM) vs. generalized metabolic growth theory (IGM). f(r) and r represent the actual tree ring growth rate and tree radius, respectively. Here, we show the best growth trajectory of trees within the same forest, so we assume that the actual f(r) (gray dots) is below the curve. It is worth noting that the IGMR-U is more common, indicating that the gray dots tend to appear more often below this curve.

**Figure 3.**Distribution of the coefficient of variation in the tree radius (

**A**), the distribution of the normalized growth rate on the relative radius (

**B**,

**C**), and the distribution of the coefficient of variation in the tree ring growth rate along the relative radius over the past 5 years. The test samples in (

**B**,

**C**) are from the data belonging to the gray and white bars in (

**A**), respectively. The green curve in (

**B**) is obtained from the 0.95th quantile fit of the data by Equation (4), where b = k = 0.736 ± 0.10. This curve, along with its half, is also plotted in (

**C**,

**D**). The red curve (A) is the normal distribution curve. (

**B**–

**D**) Dot density is represented by different colors, with warmer colors representing a higher density and cooler colors representing a lower density. The quartiles for the blue, red, and orange borders are 0.95, 0.75, and 0.55, respectively.

**Figure 4.**Correlation (

**A**), hierarchical partitioning (

**B**), and structural equation model analyses (

**C**) between explanatory variables and OVG. (

**A**) Normalized growth rate: ratio of the average growth rate of tree cores over the past five years to the average growth rate of all tree cores (f(r)

_{c}/f(r)

_{m}). (

**B**) ele: elevation; Temp: mean annual temperature; Maxtwm: max temperature of warmest month; Mentwq: mean temperature of warmest quarter; Mentcq: mean temperature of coldest quarter; Ap: annual precipitation; Pwq: precipitation of wettest quarter; Pcq: precipitation of driest quarter. (

**A**) Red line is linear regression. (

**C**) Solid red and green arrows represent significant (p < 0.05) positive and negative paths, respectively; double arrow solid lines indicate a correlation. The numbers near the lines indicate the standard path coefficients or correlation coefficients. R

^{2}represents the amount of variation of the variable explained by corresponding paths. The red and green lines indicate negative and positive effects, respectively.

**Figure 5.**Radial growth rates at the current tree radii vs. model-estimated historical best radial growth rates at the same radii (

**A**) (n = 2500), comparison of their differences at different elevations (

**B**,

**C**) (W-test, *** p < 0.001), and the correlation of this difference with the climatic factors (

**D**,

**E**). The red lines (

**D**,

**E**) are linear regression.

Species Name | Abbreviated Name | Latitude | Longitude | Average Elevation (m) | Number of Sample Sites/Tree Cores | Species Composition | Age Structure | Average ± SD/Maximum DBH (mm) | Average ± SD/Maximum Age (y) |
---|---|---|---|---|---|---|---|---|---|

Abies forestii | ABFO | 27.33–29.28 | 99.27–100.08 | 3521 | 7/345 | single species | Single/mixed age | 202 ± 94/386 | 261 ± 121/463 |

Abies recurvata | ABRC | 28.04 | 99.02 | 3200 | 1/18 | - | Single age | 276 ± 76/394 | 272 ± 88/394 |

Cupressus chengiana | CUCH | 31.78 | 101.9167 | 2500 | 1/39 | single species | - | 218 ± 73/330 | 210 ± 88/358 |

Juniperus przewalskii | JUPR | 36.00–38.57 | 97.06–99.87 | 3741 | 16/1256 | - | Single/mixed age | 162 ± 80/294 | 552 ± 288/1046 |

Juniperus tibetica | JUTI | 28.37–33.80 | 91.52–100.27 | 4136 | 12/549 | single species | Single/mixed age | 178 ± 81/323 | 407 ± 200/795 |

Picea likiangensis | PCLI | 27.58–31.95 | 96.48–100.28 | 3520 | 6/195 | - | - | 212 ± 83/341 | 232 ± 102/439 |

Tsuga dumosa | TSDU | 27.88–28.04 | 98.40–98.98 | 3125 | 2/63 | - | Single age | 310 ± 44/362 | 293 ± 82/460 |

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**MDPI and ACS Style**

Wang, Y.; Shu, S.; Wang, X.; Chen, W.
Effect of Tree Size Heterogeneity on the Overall Growth Trend of Trees in Coniferous Forests of the Tibetan Plateau. *Forests* **2023**, *14*, 1483.
https://doi.org/10.3390/f14071483

**AMA Style**

Wang Y, Shu S, Wang X, Chen W.
Effect of Tree Size Heterogeneity on the Overall Growth Trend of Trees in Coniferous Forests of the Tibetan Plateau. *Forests*. 2023; 14(7):1483.
https://doi.org/10.3390/f14071483

**Chicago/Turabian Style**

Wang, Yuelin, Shumiao Shu, Xiaodan Wang, and Wende Chen.
2023. "Effect of Tree Size Heterogeneity on the Overall Growth Trend of Trees in Coniferous Forests of the Tibetan Plateau" *Forests* 14, no. 7: 1483.
https://doi.org/10.3390/f14071483